 # Multiphase Flow in Porous Rock

Course objectives:

By taking the exam, the student will be able to mathematically describe the multiphase flow in the reservoir rock. Students will use modern methods of upscaling of data from the micro scale to the level of the core and to the level of well data and reservoirs.

Course content:

Contents of theoretical lectures

• Flow equations. Darcy, Darcy-Weisbach and Forcheimer equations
• Examples of flow calculations in linear geometry (Darcy eq.)
• Examples of flow calculations in radial geometry (Darcy eq.)
• Example of application of the equation of conservation of mass and continuity to Darcy's flow case
• Description of the nonlinear flow in the near-wellbore zone (non-Darcy flow).
• Algebraic form of a nonlinear relationship between pressure drop and flow rate.
• Principles for determining zones completely saturated with water and zones partially saturated with water from electric well-logging data
• Wettability, capillary pressure, relative permeability
• Analytical expression for capillary pressure
• Capillary effects on fluid flow. Bundle of tubes concept, effects of different pore geometries on capillary forces. Poiseuille equation.
• Buckley-Leverett (BL) theory. Basic assumptions and application. Method for determining average saturation.
• Welge - graphical interpretation of the fractional flow curve
• Numerical description of BL calculation of injection fluid front advance in time.
• Extension of the calculation of flooding by BL method, gravitational term in the calculation (inclined layer), calculation of the required amount of water for injection to a certain extraction.
• Diffuse flow - Koval's theory.
• Effects at the pore level - analysis of flow properties at the pore level
• Flow of non-Newtonian fluids in a reservoir. Application of fractional flow curve for EOR estimates.
• Formation compressibility. Stresses in the reservoir rock, and in the laboratory when measuring rock properties.
• Correlations of mechanical properties of the formation - compressibility, fracture pressure, description of the change in permeability and porosity due to the change of pressure in the reservoir (rock sample).
• Review of flow equations in differential form

Content of practical exercises

• Rock compressibility - an overview of the main parameters related to the mechanical properties of rock, basic equations and applications in reservoir engineering
• Rock compressibility - pore compressibility. Analytical tasks - correction of stock estimate based on change in pore volume by pressure reduction
• Permeability calculations - Linear and radial flow of incompressible and weakly compressible fluid (water, oil)
• Permeability calculations II - Linear and radial flow of compressible fluid (gas)
• Permeability calculations III - Permeability and flow calculations in a segmented heterogeneous model - series of permeability changes, permeability anisotropy in parallel layers
• Determination of porosity at the reservoir level - data upscaling from core-level and well-level to the level of the production/injection formation (reservoir)
• Reservoir permeability assessment - data upscaling from core-level and well-level to the level of the production/injection formation (reservoir)
• Determination of a representative capillary pressure and saturation curve above the oil-water contact
• Statistical methods for processing capillary pressures - Leverett's J (Sw) function
• Statistical processing of capillary pressure data using the cross-plot method
• Electrical resistance (estimates of porosity and saturation based on resistance data, ie formation factors)
• Acoustic properties of rock and interpretation of acoustic measurements in the laboratory
• Example of numerical calculation of flow in rock
• Program task: Multiphase calculations - Calculation of relative permeability from given laboratory results of multiphase flow, calculation of relative permeability ratios based on laboratory data of multiphase flow and Welge method.
• Program task: Calculation of multiphase immiscible flow (flooding) according to Buckley-Leverett theory

Learning outcomes at the level of the course:

• Describe mathematically physical properties of fluid flow in a porous rock and flow calculations depending on different cases of fluid flow in a porous rock;
• Match the laboratory and well-logging data;
• Formulate the data needed for waterflooding calculations and other fluid injection;
• Formulate the inputs for reservoir simulation models.

## Pre-register

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